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Journal of Applied Science and Engineering, Vol. 15, No. 1, pp. 89-96 (2012)
89
The Irrelevant Values Problem of Decision Tree in
Medical Examination
Kuang-Yi Chou3, Huan-Chao Keh1, Nan-Ching Huang1*, Shing-Hwa Lu2,
Ding-An Chiang1 and Yuan-Cheng Cheng1
1
Department of CSIE, Tamkang University,
Tamsui, Taiwan 251, R.O.C.
2
Department of Urology, School of Medicine, National Yang-Ming University and Department of Urology,
Zhong Xiao Branch, Taipei City Hospital, Urological Center, Taipei, Taiwan 115, R.O.C.
3
Center of General Education, National Taipei University of Nursing and Health Sciences,
Taipei, Taiwan 112, R.O.C.
Abstract
Data mining technique is extensively used in medical application. One of key tools is the
decision tree. When a decision tree is represented by a collection of rules, the antecedents of individual
rules may contain irrelevant values problem. When we use this complete set of rules to medical
examinations, the irrelevant values problem may cause unnecessary economic burden both to the
patient and the society. We used a hypothyroid disease as an example for the study of irrelevant values
problem of decision tree in medical examination. Hypothyroid disease is used to associate to the
mechanism of thyroid-stimulating hormone (TSH). Physicians will combine lots of information; such
as patient’s clinical records, medical images, and symptoms, prior to the final diagnosis and treatment,
especially surgical operation. Therefore, to avoid generating rules with irrelevant values problem, we
propose a new algorithm to remove irrelevant values problem of rules in the process of converting the
decision tree to rules utilizing information already present in the decision tree. Our algorithm is able to
handle both discrete and continuous values.
Key Words: Decision Tree, Classification, Irrelevant Values, Missing Branches, Medical Examination
1. Introduction
The decision tree is based on the application of recursive formula and the algorithm of top-down and divide-and-conquer. The irrelevant values problem may be
generated during the construction of decision tree. The
structure of the decision tree may be altered at the cut-off
point while irrelevant values are removed. On the other
hand, the decision tree can handles continuous values
through a local discretization which the dependency
among all attributes must relate to each other; if the
dependency does not exist, a global discretization is a
better choice.
*Corresponding author. E-mail: [email protected]
The ID3 algorithm is commonly brought to solve the
irrelevant values problem of a decision tree. When a decision tree does not have abundance of data, the grafting
technique is mostly applied to its algorithm. As a result, a
best test result can be obtained from a cut-off point where
its irrelevant values problem has been removed. In this
paper, it proposes an algorithm which removes irrelevant
values problem and uses association rules to integrate
both classification rules and a cut-off point with global
discrete attributes. At same time, it creates a classifier to
integrate all classification rules. A decision tree, moreover, is built up by selecting the best test attribute as the
root of the decision tree. Then, the same procedure is
operated on each branch to lead on the remaining levels
of the decision tree until all examples in a leaf belong to
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the same class. The decision tree; however, creates a
branch for each value which appears in the training data
without considering whether the value is relevant to the
classification. In addition the resultant tree may have the
over-specialization problems [1-3]. Without losing generality, we only consider ID3-like algorithm in our
study.
Cheng et al. [4] and Bohanec and Rajkovic [5]
pointed out that the irrelevant values problem and the
missing branches problem are two major causes of overspecialization in the decision tree. The missing branches
problem is due to the reduced subsets at the non-leaf
nodes which do not necessarily contain examples of
every possible value of the branching attribute. Consequently, the decision tree may fail to classify some instances. Since some values of that attribute may be irrelevant to the classification; the resultant rules of the
decision tree may have irrelevant conditions, which extra information must be supplied [5,6]. Extra information
means more examinations and costs are required to the
patient and the society. When the decision tree is applied
to medical applications we have to deal with the irrelevant conditions in the decision tree to save medical resources and avoid unnecessary examinations.
For example, let us consider the decision tree in Figure 1; we are going to focus on the irrelevant values in
the branch Br. As shown in Figure 1, branches Br1 and
Br2 can be represented by a1 Ù b1 ® c1 and a2 Ù b1 ® c1;
however, values a1 and a2 are irrelevant to each other respectively. Consequently, these rules can be generalized
by deleting these irrelevant values without affecting its
accuracy; leaving the more appealing rule, b1 ® c1. The
resultant rule is more concise and comprehensible than
the original rule. However, its drawback is likely to suffer from missing value problem. This kind of rules may
be useful in many applications. For example, the doctor
can examine item B first to make sure whether or not this
patient needs to take item A. This process can reduce
some burdens; for example, expense, inconvenience or
harm to the patient.
Fayyad proposed two algorithms, GID3 and GID3*,
to solve the over-specialization problem of the decision
tree constructed using ID3 [4,7-9]. Both algorithms
over-whelm the irrelevant values problem at attribute
phantomization step before attribute selection. However,
the problems found in branches of GID3 and of GID3* is
longer than the branches found in ID3.
Quinlan pointed out that the tree is represented by a
collection of rules and each leaf in the tree would not result in much simpler than the tree which one rule is for
every leaf [6,10]. Therefore, Quinlan used a pessimistic
estimation on the rule accuracy to generalize each rule
by removing not only irrelevant conditions but also conditions that hardly discriminated nominated class from
other classes [6,11]. Since the antecedents of a rule may
contain irrelevant conditions, the deletion of superfluous
conditions is result in a generalized rule regardless of its
correctness and is simpler than the tree. Jerez-Aragones
et al. [1] and Chiang et al. [12] provided another view for
the solution of over-specialization problem from a decision tress which its attributes contain discrete values
only. They eliminated irrelevant values in the process of
converting the decision tree to the rules for classification
according to the information on the decision tree. Our
algorithm does not use example cases to guide the conversion process. It has clear computational advantages
which can be integrated into any existing decision tree
system easily. The empirical results present evidences
that our algorithm overcomes not only the irrelevant values problem, but also the missing branch problems with
negligible cost of concerning the construction of decision tree [13-15]. Consequently, the average error rate of
the decision tree is also reduced. The new algorithm can
not only deal with discrete values but also with continuous ones.
The decision tree is one of the key data mining techniques in medical application. As a matter of fact, decision tree of nowadays does not have a specific system for
certain types of data, because the algorithms generated
from a decision tree are varied and depends on type of
Figure 1. A decision tree with irrelevant values.
The Irrelevant Values Problem of Decision Tree in Medical Examination
input data. Because of digitization of medial information, any diagnosis has to correlate with various medial
records from different database and the process may
cause unnecessary economic burden both to the patient
and the society. We used hypothyroid disease which is
related to the mechanism of thyroid-stimulating hormone (TSH) as a clinical example for the study of irrelevant values problem in the decision tree. The attributes
of many medical data are continuous values; therefore,
the new algorithm may be used to solve irrelevant values
problem of a decision tree
An algorithm for identifying irrelevant values problem of the decision tree with discrete values is introduced in section 2. The new algorithm for removing irrelevant values of the decision tree with continuous values is given in section 3. Since the discrete values can be
viewed as a special case of the continuous values; as a
result, the new algorithm can deal with both discrete and
continuous values. The irrelevant values problem for
medical examination is discussed in section 4. The conclusions are stated in section 5.
2. An Algorithm to Identify Irrelevant Values
to Discrete Values
In this section, we introduced some definitions and
theorems from our previous work with respect to identify
irrelevant values of the decision tree with discrete values
[12]. Let A = {A1, ..., An} be a set of attributes, C = {C1,
..., Cs} be a set of classes, and a = (aij, ..., anm) be a
branch’s values of Br. To represent a decision tree by a
set of branches, the branch Br of the decision tree can be
represented as the form, Br[A1] Ù ... Ù Br[An] ® Ck or a1r
Ù ... Ù anm ® Ck, where Br[Ai] is the branch value out of
an attribute Ai in the branch Br, aij Î Br[Ai], i = 1 … n and
1 £ k £ s.
For the decision tree, not all attributes Ai will be the
nodes of branches Br in the decision tree. When the corresponding values of Br[Ai] are missing, these attributes
are irrelevant with respect to Br. If there are many rules
implied by Br, the irrelevant attributes will be considered. This observation can be explained by the following definitions.
Definition 1.
Let Br[A1] Ù ... Ù Br[Aj-1] ® Ck be a branch in the de-
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cision tree, then the rules with respect to attributes A1, ...,
An implied by Br are:
{Br[A1] Ù ... Ù Br[Aj-1] Ù aj Ù ... Ù ans ® Ck | ajr , ..., ans Î
domain (Aj, ..., An)},
where domain (Aj, ..., An) = domain (Aj) ´ … ´ domain (An).
To easily identify irrelevant values of a branch in a
decision tree, we further define the following definition.
Definition 2.
Let Br and Br¢ be two different branches in a decision tree, where Br = Br[A1] Ù ... Ù Br[Aj] ® Ck1, then Br
is in conflict with Br¢ with respect to attributes A1 … Aj if
and only if Br[A1] Ù ... Ù Br[Aj] ® Ck2 is a part of rule implied by Br¢ and Ck1 ¹ Ck2.
According to the semantics of irrelevant values; a
value, Br[Aj], is an irrelevance of a rule. This value can
be deleted or replaced by any value from the same domain value without affecting the correctness of the rule.
Therefore, based on definitions 1 and 2, a combinatorial
explosion can be applied to the number of comparisons
to all the branches for identification of irrelevant values
of a branch. The process of identifying irrelevant values
by definition 1 and 2 is very time-consuming. To enable
users to focus on only relevant conditions of the rules,
the following theorems are able to work out the irrelevant values problem for a complex decision tree. These
theorems were proven in [13,14,16].
Theorem 1.
Let Br[A1] Ù ... Ù Br[Aj-1] Ù Br[Aj] Ù Br[Aj+1] Ù ... Ù
Br[An] ® Ck be a branch through a non-leaf node P in a
decision tree, and the branching attribute with respect to
P be Aj. For all branches through P of the decision tree, if
Br is not in conflict with these branches with respect to
attributes Aj+1 ... An, then Br[Aj] is an irrelevant value in
Br.
Theorem 2.
Let Br be a branch through a non-leaf node P of the
decision tree. When the branch value Br[P] has been
identified by theorem 1. Other branches through P are
useless for the following process to identify the irrelevant values of Br.
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Kuang-Yi Chou et al.
Theorem 3.
Let Br[A1] Ù ... Ù Br[Aj] Ù Br[Aj¢] Ù ... Ù Br[An1] ®
Ck1 and Br¢ [A1] Ù ... Ù Br¢[Aj] Ù ... Ù Br¢[An2] ® Ck2 be
two branches through a non-leaf node P in the tree,
where the branching attribute with respect to P is Aj. Let
A = {Aj¢, ..., An1} and A1 be the same attributes in these
two branches, where A1 Í A then, Br is in conflict with
Br¢ with respect to A if and only if Br[A1] = Br¢[A1] and
Ck1 ¹ Ck2.
According to theorem 1, for each selected node P,
the branch value of node P can identify whether the
branch value of node P is an irrelevant value of a branch,
Br, or not. By theorem 2, if the branch value of node P is
an irrelevant value, then branches, which are through
node P, can be ignored for the following process to identify irrelevant values of Br. In other words, by theorem 1
and theorem 2, to identify all the irrelevant values of a
branch, we need to check all the branches in the decision
tree only once. Moreover, since we do not have to consider the rules implied by each branch in the decision tree
by theorem 3, the computation time of identifying whether two branches are conflict between each other can be
reduced greatly. Actually, without losing generality,
since the number of common nodes of two branches is
always small, we can assume that the time complexity of
identifying whether two branches are in conflict with
each other is constant. Therefore, the time complexity of
identifying all irrelevant values of a branch by these theorems is reduced to O(m) at worst case, where m is the
number of branches of the tree.
3. An Algorithm to Identify Irrelevant Values
to Continuous Values
For many applications, some attributes may contain
continuous values. Therefore, we provided a new algorithm, which is represented by the following theorem, to
solve the irrelevant values problem of the decision tree
with continuous values.
Theorem 4.
Let Br[A1] Ù ... Ù Br[Aj] Ù Br[Aj¢] Ù ... Ù Br[An1] ®
Ck1 and Br¢[A1] Ù ... Ù Br¢[Aj] Ù ... Ù Br¢[An2] ® Ck2 be
two branches through a non-leaf node P in the tree,
where the branching attribute with respect to P is Aj. Let
A = {Aj¢, ..., An1}, A1 be the same attributes in these two
branches and a1 be a branch’s values of Br[A1], where A1
Í A. Then, Br is in conflict with Br¢ with respect to A if
and only if
when A1 ¹ Æ, $ a1, a1 Î Br[A1], a1 Î Br¢[A1] and Ck1 ¹
Ck2 (1) or
when A1 = Æ, Ck1 ¹ Ck2 (2).
Proof.
Let A = {Aj¢, ..., An1} and A1 be the same attributes in
these two branches.
Let A1 ¹ Æ, a1 be the branch’s values, and a1 Î
Br[A1] (1). When " a1, a1 Ï Br¢[A1], it implies that these
two branches will never be in conflict with each other
with respect to A1 by definition 2. Therefore, we need
only to consider the case $ a1, a1 Î Br[A1], a1 Î Br¢[A1]
and Ck1 ¹ Ck2. According to definition 1, when a1 Î
Br¢[A1], a1 ® Ck2 must be a part of rule implied by Br¢.
Therefore, Br must be in conflict with Br¢ with respect to
A if and only if $ a1, a1 Î Br[A1], a1 Î Br¢[A1] and Ck1 ¹
Ck2 by definition 2.
Let A1 = Æ and Ck1 ¹ Ck2 (2). According to definition
1, when A1 = Æ, it implies that " a, a ® Ck2 must be a
part of rule which is implied by Br¢, where a Î Br[A].
Since Ck1 ¹ Ck2, Br must be in conflict with Br¢ with respect to A by definition 2.
According to theorem 4, when Ck1 = Ck2; branches Br
® Ck1 and Br¢ ® Ck2, are never in conflict with each
other. To identify all irrelevant values of a branch Br, we
need only to consider those branches; Br¢, whose leaves
are different from Ck1. When attribute has only discrete
and finite values, it can check easily whether it has
branch values in Br and Br¢ at the same time or not based
on theorem 3. On the other hand, when some attributes
contain continuous values, the situation becomes more
complicated. When the branching attribute, Br[A], contains continuous values, the branch value of A is computed from the training data and it may or may not appear
in the training data. After the branch’s value of A is determined, the values of A in Br is ranged by one of £, >, <
or ³ function. Consequently, we have to check the training data; otherwise, it is very hard to known whether or
not Br is in conflict with Br¢ by theorem 4.
For example, let Br = A > 5.8 ® c1, Br¢ = A £ 6 ® c2,
and c1 ¹ c2. According to theorem 4 and without consideration of the training data, we will conclude that Br
The Irrelevant Values Problem of Decision Tree in Medical Examination
is in conflict with Br¢ because Br[A] È Br¢[A] ¹ Æ. However, since the braches values of continuous values is
computed from training data, the value 5.8 may not be in
the range of Br[A]. In other words, if the smallest value
in the range of Br[A] is larger than 6, the new conclusion
is that Br is not in conflict with Br¢. Therefore, we have
to check the training data in the process of removing all
irrelevant conditions. Otherwise, some irrelevant values
can not be removed. Even that, our algorithm is still useful to integrate into any existing decision tree system
without training data of example 3. In addition, we can
not modify the existing systems and can not use example
cases to guide the conversion process. To solve this problem, we may scan the database one time for the continuous value attributes to find out the largest and smallest values from the corresponding training data. Consequently, all irrelevant values were removed from the resultant rules without usage of training data. Moreover, to
make the process of removing all irrelevant values from
a branch more efficient, we recursively applied the
above theorems until the node P was at the root of decision tree. The corresponding algorithm is shown below.
Input: A decision tree
Output: A set of rules without irrelevant conditions;
Let Br = {Br1, ..., Brm}; /* the branches of the decision
tree */
For each branch Br¢ in Br Do
{Let Br¢ = Br¢[A1] Ù ... Ù Br¢[Ak] ® Ci
For j = k down to 1 Do
{Apply theorem 1, 2 and 4, and check whether Br¢[Aj] is
an irrelevant value;
If Br¢[Aj] is an irrelevant value Then remove Br¢[Aj]
from Br¢;}
Represented Br¢ by a rule;}
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our decision tree study. One of most interesting cases is
hypothyroid disease which is related to the mechanism
of thyroid-stimulating hormone (TSH). TSH is secreted
by thyrotrope cells in the pituitary gland, which regulates
the endocrine function of the thyroid gland. TSH stimulates the thyroid gland to secrete the triiodothyronine
(T3) and thyroxine (T4). The level of T3 and T4 in the
blood has an effect on the pituitary release of TSH; when
the levels of T3 and T4 are low, the production of TSH is
increased, and on the converse, when levels of T3 and T4
are high, TSH production is decreased. This effect creates a regulatory negative feedback loop. Additionally,
the level of T3 and T4 in the blood is the index for diagnosis of thyroid-related disease.
As a result, we use the decision tree depicted in Figure 2 to discuss the irrelevant values problem regarding
to medical examinations of hypothyroid disease. The
rules found in Table 1 have two conditions; one is irrelevant and the other is no irrelevant. When two conditions are compared to each other, some characters can be
found.
First, the decision tree model recommends the doctors involve many unnecessary investigations for the
Let us consider the decision tree depicted in Figure
2. This decision tree was proposed by Quinlan for hypothyroid disease examinations. Without consideration of
training data in the database, the original rules and the resultant rules are shown in Table 1. Comparing these two
sets of rules, we found out that our rules are more concise
and comprehensible than the original rules.
4. Our Algorithm for Medical Examinations
There are many medical examinations available for
Figure 2. The ID3 decision tree for the diagnosis of hypothyroid disease (unit in mIU/L).
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Kuang-Yi Chou et al.
Table 1. The recurred rules of the decision tree (unit in mIU/L)
The original rules without considering irrelevant values problem
The resultant rules without irrelevant conditions
FTI £ 64 Ù TSH measured = f ® Negative
FTI £ 64* Ù TSH measured = t* Ù TSH £ 5.8 ® Negative
FTI £ 64 Ù TSH measured = t Ù TSH > 5.8 Ù Thyroid surgery = f
® Primary hypothyroid
FTI £ 64 Ù TSH measured = t Ù TSH > 5.8* Ù Thyroid surgery = t
® Negative
FTI > 64 Ù TSH £ 6 ® Negative
FTI > 64 Ù TSH > 6* Ù On thyroxin = t ® Negative
FTI > 64 Ù TSH > 6 Ù On thyroxin = f* Ù TSH measured = f ®
Negative
FTI > 64 Ù TSH > 6 Ù On thyroxin = f Ù TSH measured = t Ù
Thyroid surgery ® Negative
FTI > 64 Ù TSH > 6 Ù TSH measured = t Ù Thyroid surgery = f* Ù
TT4 > 150 ® Negative
FTI > 64 Ù TSH > 6 Ù TSH measured = t Ù thyroid surgery = f Ù
TT4 £ 150 Ù TT4 measured = f ® Primary hypothyroid
(1) TSH measured = f ® Negative
(2) TSH £ 5.8 ® Negative
(3) FTI £ 64 Ù THS measured = t Ù TSH > 5.8 Ù
Thyroid surgery = f ® Primary hypothyroid
(4) Thyroid surgery = t ® Negative
*
FTI > 64 Ù TSH > 6 Ù THS measured = t Ù Thyroid surgery = f Ù
TT4 £ 150 Ù TT4 measured = t ® Compensated hypothyroid
(5) FTI > 64 Ù TSH £ 6 ® Negative
(6) FTI > 64 Ù On thyroxin = t ® Negative
The resultant rule is equal to (1)
The resultant rule is equal to (4)
(7) FTI > 64 Ù TT4 > 150 ® Negative
(8) TSH > 6 Ù THS measured = t Ù Thyroid
surgery = f Ù TT4 £ 150 Ù TT4 measured = f
® Primary hypothyroid
(9) FTI > 64 Ù TSH > 6 Ù THS measured = t Ù
Thyroid surgery = f Ù TT4 £ 150 Ù TT4
measured = t ® Compensated hypothyroid
FTI £ 64*: Free thyroxin (FT4) can be measured or calculated directly as the free thyroxin index (FTI). The FTI is a
level of T4 in relation to the amount of thyroxin-binding globulin present. The FTI is calculated from the T4 and T3
uptake, or T3U. The FTI value can indicate when an abnormal level of thyroxin-binding globulin in the blood causes an
abnormal level of T4. Also the FTI can help tell if abnormal amounts of T4 are present because of abnormal amounts of
thyroxin-binding globulin. TSH > 5.8* and TSH > 6: The interaction between feedback mechanism and thyroid relies
on TSH value. Normal values range from 0.4 to 5.8 mIU/L for people with no symptoms of an under-active thyroid.
Values below 0.4 are considered hyperthyroid; values 5.8 mIU/L or slightly higher are considered sub clinical
hypothyroid, and values generally above 10 mIU/L indicate full hypothyroid condition. TSH measured = t*: TSH
measured is a qualitative measurement of TSH. True indicates an abnormal plasma TSH level and False refers to a
normal level of TSH. Thyroid surgery = f*: This means a patient without thyroid surgery. Thyroid surgery could result
in the low level of thyroxin which mimics a manifestation of hypothyroid. On thyroxin = f*: Patient who is under
medication of thyroxin will change the level of serum thyroxin. False result indicates the patients are not under thyroxin
medication. The Table 2 is a comparison of different types of investigation diagnostic methods.
patients. For example, the rules of 1, 2, and 4 show that
they are much simpler for physicians to make diagnosis
from the investigation results. In our algorithm, a history
with negative TSH, thyroid surgery, or TSH less than 5.8
leads the diagnosis to a negative result. This means that
if a patient has a history of thyroid surgery or thyroxin
medication, his diagnosis will always be negative and
doe not need to have the other test like TT4 or FT4. For a
patient with a measurement of TSH below 5.8, the aforementioned tests are not required. Only when TSH is
above 5.8; patient will need the examination of FTI or
TT4 to make a correct diagnosis.
Second, the proposed algorithm shows a better way
for a physician to make a correct diagnosis without prob-
Table 2. Features of the types of investigation in diagnosis
of hypothyroid disease
Types of
investigation
TSH
TSH measured
Thyroid surgery
TT4
On thyroxin
FTI
Cost
Convenience
Harmful
Medium
Low
Low
Medium
Low
High
Yes
Yes
Yes
Yes
Yes
Yes
No
No
No
No
No
No
ing every available investigation. Only the patients without a thyroid surgery history and thyroxin medication
need to have their FTI, TT4 and TSH examined to make
The Irrelevant Values Problem of Decision Tree in Medical Examination
the diagnosis. On the contrary the decision tree model
shows that all tests are needed. Therefore, we can optimize lab investigations for diagnosing a possible hypothyroid patient with a new algorithm than the original decision model. Less investigation is both good for patients
and the society. For each patient, he/she might not spend
time and money to the medical examination which is not
needed for diagnosis. For our society, it is a waste if we
spend our fiscal expenditure for any medical procedure
that is not quite needed. The doctors’ time is limited, the
patients are suffering from the disease both physically
and mentally, the ever-increasing cost of medical expenditure is a huge burden to the governments. So it is a
great benefit for all parts if we can find alternatives to cut
down the medical expenditure, such as the lab investigations for the diagnosis.
Third, the decision tree model shows that different
values in TSH, FTI, T4, and TT4 could lead to different
diagnosis which depends on the value of other investigations. When combination of numerous investigations is
involved, physicians could make mistakes easily with a
tight schedule. The new algorithm reduces the irrelevant
rules and makes the diagnosis much simpler and; therefore, it is less likely to make mistakes in clinical work.
Therefore, the new algorithm model cuts down the rules
and helps clinicians to find the easiest and fastest way to
apply suitable investigations for a particular patient
correctly.
5. Conclusion
The solution of removing irrelevant values problem
depends only on the semantics of the decision. Consequently, our new algorithm can overcome the irrelevant
values problem by integration of existing tree-construction algorithm without increase of computational cost for
the construction of decision tree. Medical problem is a
field where “decision tree” algorithm is usually applied.
As the cost of healthcare system is rising again and
again, it is urgent for government to reduce the fast increase of the medical expenditure while maintaining the
quality. Using the decision tree model could be helpful
for us to retrieve the relevant medical tests needed in
clinical cases under different condition. However, the
new rules we adopted here and testified in a hypothyroid
model showed more efficient and more useful to exclude
95
the irrelevant conditions. Therefore, the new rules could
be a good technique for physicians to find the easiest and
fastest way to make a diagnosis and cut down the indispensable expenditures of lab investigations. Moreover,
as discussed in section 3, the rules may be further reduced by considering the training data. We pause to collect real data to do the cases. Therefore, we plan to combine these topics into our algorithm in the near future.
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Manuscript Received: Dec. 2, 2010
Accepted: Jun. 20, 2011